As shown in Sec. , the spin operator \(\vec{S}\)  and exciton operator \(\vec{P}\)  can be expressed either with bosonic partons \(b_{a\sigma }\)  or fermionic parton \(f_{a\sigma }\) . When the layer polarized Mott insulator is in the magnetically ordered phase, the natural parton description should be the
Schwinger boson theory shown in Sec. . When \(t/U\)  is in the intermediate regime, the layer polarized Mott insulator may be in a spinon Fermi surface state[1]}, [2]}. The exact nature of the intermediate weak Mott insulator of the spin 1/2 Hubbard model is still under debate and may depend on details like whether further neighbor hoppings are included or not. In the experimental study of the metal-insulator transition in the AA stacked MoTe\(_2\) /WSe\(_2\)  moiré system[3]}, no magnetic order is found down to the lowest temperature. The spin susceptibility in the weak Mott regime seems to be just like the metallic phase, suggesting a possible spinon Fermi surface ground state. However, such a neutral spinon Fermi surface is hard to detect even if it already exists in the MoTe\(_2\) /Wse\(_2\)  moiré system. In this section we propose to study a MoTe\(_2\) -hBN-MoTe\(_2\) /WSe\(_2\)  system similar to the moiré+monolayer setting up in Ref. gu2021dipolar,zhang2021correlated. If the layer polarized Mott insulator indeed hosts a spinon Fermi surface, then at finite exciton density \(x>0\) , the most natural phase is a \(U(1)\)  spin liquid with spinon Fermi surfaces in both layers. Especially the neutral fermion in the top layer can be viewed as a fermionic exciton formed by electron \(c_{t\sigma }^\dagger \)  in the top layer bound to the holon \(\varphi \)  in the bottom layer, as illustrated in Fig. REF . Possibility of fermionic excitons has also been discussed previously in Kondo insulator SmB\(_6\)[4]} or in bilayer Landau levels[5]}, [6]}, but in our realization the neutral fermion carries a charge under a \(U(1)\)  symmetry generated by \(P_z\) . This makes it possible to detect the movement of the neutral fermionic excitons through counter-flow transport, which has not been pointed out in these previous examples.
